cp's OEIS Frontend

A308612 Moments of the ternary Cantor measure (numerators).

%I A308612 #43 Nov 19 2023 15:56:11
%S A308612 1,1,3,5,87,31,10215,2675,2030721,3791353,21043039755,3617048975,
%T A308612 456510966890031,66989072882759,4380916942893971361,
%U A308612 8410761713598485675,4355410489470724905492213,2471070675390828342358441,8410576873515817323688553597445
%N A308612 Moments of the ternary Cantor measure (numerators).
%C A308612 The ternary Cantor measure, defined many ways, is the unique Borel measure mu on the unit interval [0,1] satisfying the following recurrence relation for any measurable set E: mu(E) = mu(phi_0(E))/2 + mu(phi_2(E))/2.  Here, for j in {0,1,2}, phi_j: [0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/3.  For any nonnegative integer k, we define the k-th moment to be I_k to be the integral of x^k with respect to mu.  The described sequence I(0), I(1), I(2), ... is rational and this sequence a(0), a(1), a(2), ... is the sequence of numerators of I(0), I(1), I(2), ....
%C A308612 For the purpose of computing I_k, we note the following recurrence relation: I(0) = 1 and for all positive k, I(k) = (1/(3^k-1))*((1/2) * Sum_{j=0..k-1} binomial(k, j) + (1/2) * Sum_{j=0..k-1} binomial(k, j) * 2^(k-j) * I(j)).
%C A308612 More generally, for any N-dimensional nonnegative vector alpha = (alpha_0, ..., alpha_{N-1}) whose entries sum to 1, there exists a unique Borel measure mu = mu^{alpha} on [0,1] so that for any measurable set E, the following identity holds: mu(E) = Sum_{k=0..N-1} alpha_k * mu(phi_k(E)).  Here, for j in {0, 1, ..., N-1}, phi_j: [0,1] to [0,1] is the linear function which sends x in [0,1] to (x+j)/N.  Defining I(k) to be the integral of x^k with respect to mu, the following recurrence relation holds: I(0) = 1 and for all positive k, I(k) = (1/(N^k-1)) * Sum_{n=0..N-1} alpha_n * Sum_{j=0..k-1} binomial(k, j) * n^{k-j}*I(j).
%H A308612 Michael De Vlieger, <a href="/A308612/b308612.txt">Table of n, a(n) for n = 0..97</a>
%H A308612 David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Michael G. Rose, <a href="https://doi.org/10.1016/j.amc.2013.06.078">Expectations on fractal sets</a>, Applied Mathematics and Computation, Vol. 220 (2013), pp. 695-721, <a href="https://www.davidhbailey.com/dhbpapers/fracboxes.pdf">alternative link</a>.
%H A308612 Steven N. Harding, Alexander W. N. Riasanovsky, <a href="https://arxiv.org/abs/1908.05358">Moments of the weighted Cantor measures</a>, arXiv:1908.05358 [math.FA], 2019.
%H A308612 Math Stack Exchange, <a href="https://math.stackexchange.com/q/57721">Integrating f(x) = x for x in C, the Cantor set, with respect to a certain measure</a>
%t A308612 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, j]*2^(n - j - 1)*a[j], {j, 0, n - 1}]/(3^n - 1); Table[Numerator[a[i]], {i, 0, 19}] (* _Amiram Eldar_, Aug 03 2019 *)
%o A308612 (Sage)
%o A308612 def am(m, alpha):
%o A308612     N = len(alpha)
%o A308612     am = [1]
%o A308612     for a in [1..m]:
%o A308612         mm = 0
%o A308612         for k in [0..N-1]:
%o A308612             for r in [0..a-1]:
%o A308612                 mm += alpha[k]*binomial(a, r)*k^(a-r)*am[r]
%o A308612         mm /= (N^a-1)
%o A308612         am.append(mm)
%o A308612     return am
%o A308612 [p.numerator() for p in am(15, (1/2, 0, 1/2))]
%Y A308612 Cf. A308613 (denominators).
%K A308612 nonn,frac
%O A308612 0,3
%A A308612 _Alexander Riasanovsky_, Jun 10 2019
%E A308612 More terms from _Amiram Eldar_, Aug 03 2019